Modification of fourier approximation for solving boundary value problems having singularities of boundary layer type

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Modification of fourier approximation for solving boundary value problems having singularities of boundary layer type. / Semisalov, Boris; Kuzmin, Georgy.

In: CEUR Workshop Proceedings, Vol. 1839, 01.01.2017, p. 406-422.

Research output: Contribution to journal › Conference article › peer-review

BibTeX

@article{1cb1d06a57fe408fa37557d1d7a08022,

title = "Modification of fourier approximation for solving boundary value problems having singularities of boundary layer type",

abstract = "A method for approximating smooth functions has been developed using non-polynomial basis obtained by mapping of Fourier series domain to the segment [1, 1]. High rate of convergence and stability of the method is justified theoretically for four types of coordinate mappings, the dependencies of approximation error on values of derivatives of approximated functions are obtained. Algorithms for expanding of functions into series with coupled basis composed of Chebyshev polynomials and designed non-polynomial functions are implemented. It was shown that for functions having high order of smoothness and extremely steep gradients in the vicinity of bounds of segment the accuracy of proposed method cardinally exceeds that of Chebyshev's approximation. For such functions method allows to reach an acceptable accuracy using only = 10 basis elements (relative error does not exceed 1 per cent).",

keywords = "Boundary value problem, Chebyshev polynomial, Collocation method, Coordinate mapping, Estimate of convergence rate, Fourier series, Non-polynomial basis, Singular perturbation, Small parameter",

author = "Boris Semisalov and Georgy Kuzmin",

year = "2017",

month = jan,

day = "1",

language = "English",

volume = "1839",

pages = "406--422",

journal = "CEUR Workshop Proceedings",

issn = "1613-0073",

publisher = "CEUR-WS",

}

RIS

TY - JOUR

T1 - Modification of fourier approximation for solving boundary value problems having singularities of boundary layer type

AU - Semisalov, Boris

AU - Kuzmin, Georgy

PY - 2017/1/1

Y1 - 2017/1/1

N2 - A method for approximating smooth functions has been developed using non-polynomial basis obtained by mapping of Fourier series domain to the segment [1, 1]. High rate of convergence and stability of the method is justified theoretically for four types of coordinate mappings, the dependencies of approximation error on values of derivatives of approximated functions are obtained. Algorithms for expanding of functions into series with coupled basis composed of Chebyshev polynomials and designed non-polynomial functions are implemented. It was shown that for functions having high order of smoothness and extremely steep gradients in the vicinity of bounds of segment the accuracy of proposed method cardinally exceeds that of Chebyshev's approximation. For such functions method allows to reach an acceptable accuracy using only = 10 basis elements (relative error does not exceed 1 per cent).

AB - A method for approximating smooth functions has been developed using non-polynomial basis obtained by mapping of Fourier series domain to the segment [1, 1]. High rate of convergence and stability of the method is justified theoretically for four types of coordinate mappings, the dependencies of approximation error on values of derivatives of approximated functions are obtained. Algorithms for expanding of functions into series with coupled basis composed of Chebyshev polynomials and designed non-polynomial functions are implemented. It was shown that for functions having high order of smoothness and extremely steep gradients in the vicinity of bounds of segment the accuracy of proposed method cardinally exceeds that of Chebyshev's approximation. For such functions method allows to reach an acceptable accuracy using only = 10 basis elements (relative error does not exceed 1 per cent).

KW - Boundary value problem

KW - Chebyshev polynomial

KW - Collocation method

KW - Coordinate mapping

KW - Estimate of convergence rate

KW - Fourier series

KW - Non-polynomial basis

KW - Singular perturbation

KW - Small parameter

UR - http://www.scopus.com/inward/record.url?scp=85020513805&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:85020513805

VL - 1839

SP - 406

EP - 422

JO - CEUR Workshop Proceedings

JF - CEUR Workshop Proceedings

SN - 1613-0073

ER -

ID: 10186309